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10
Pure Type Systems with Definitions
, 1993
"... In this paper, an extension of Pure Type Systems (PTS's) with definitions is presented. We prove this extension preserves many of the properties of PTS's. The main result is a proof that for many PTS's, including the Calculus of Constructions, this extension preserves strong normalisation. ..."
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Cited by 20 (1 self)
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In this paper, an extension of Pure Type Systems (PTS's) with definitions is presented. We prove this extension preserves many of the properties of PTS's. The main result is a proof that for many PTS's, including the Calculus of Constructions, this extension preserves strong normalisation.
Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Cited by 13 (6 self)
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction in calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from terms (when possible), and with perpetual red ..."
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Cited by 7 (0 self)
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This paper surveys a part of the theory of fireduction in calculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from terms (when possible), and with perpetual redexes, i.e., redexes whose contraction in terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in calculus and type theory. 1. Introduction Considerable attention has been devoted to classification of reduction strategies in typefree calculus [4, 6, 7, 15, 38, 44, 81]see also [2, Ch. 13]. We are concerned with strategies differing in the length of reduction paths. This paper draws on several sources. In late 1994, van Raamsdonk and Severi [59] and Srensen [66, 67] independently developed ...
Perpetual Reductions in λCalculus
, 1999
"... This paper surveys a part of the theory of fireduction in λcalculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λterms (when possible), and with perpetual r ..."
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Cited by 6 (0 self)
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This paper surveys a part of the theory of fireduction in λcalculus which might aptly be called perpetual reductions. The theory is concerned with perpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from λterms (when possible), and with perpetual redexes, i.e., redexes whose contraction in λterms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in λcalculus and type theory.
Effective Longest and Infinite Reduction Paths in Untyped λCalculi
, 1996
"... A maximal reduction strategy in untyped λcalculus computes for a term a longest (finite or infinite) reduction path. Some types of reduction strategies in untyped λcalculus have been studied, but maximal strategies have received less attention. We give a systematic study of maximal strategies, rec ..."
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Cited by 5 (2 self)
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A maximal reduction strategy in untyped λcalculus computes for a term a longest (finite or infinite) reduction path. Some types of reduction strategies in untyped λcalculus have been studied, but maximal strategies have received less attention. We give a systematic study of maximal strategies, recasting the few known results in our framework and giving a number of new results, the most important of which is an effective maximal strategy in fij. We also present a number of applications illustrating the relevance and usefulness of maximal strategies.
A NOTE ON SHORTEST DEVELOPMENTS
, 708
"... Abstract. De Vrijer has presented a proof of the finite developments theorem which, in addition to showing that all developments are finite, gives an effective reduction strategy computing longest developments as well as a simple formula computing their length. We show that by applying a rather simp ..."
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Abstract. De Vrijer has presented a proof of the finite developments theorem which, in addition to showing that all developments are finite, gives an effective reduction strategy computing longest developments as well as a simple formula computing their length. We show that by applying a rather simple and intuitive principle of duality to de Vrijer’s approach one arrives at a proof that some developments are finite which in addition yields an effective reduction strategy computing shortest developments as well as a simple formula computing their length. The duality fails for general βreduction. Our results simplify previous work by Khasidashvili. 1.
CsR9545 1995
 Centrum voor Wiskunde en Informatica (CWI
, 1995
"... Using a characterisation of strongly normalising terms, we give new and simple proofs of the following: 1. all developments and superdevelopments are finite, 2. a certain rewrite strategy is perpetual, 3. a certain rewrite strategy is maximal and thus perpetual, 4. simply typed calculus is strongl ..."
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Using a characterisation of strongly normalising terms, we give new and simple proofs of the following: 1. all developments and superdevelopments are finite, 2. a certain rewrite strategy is perpetual, 3. a certain rewrite strategy is maximal and thus perpetual, 4. simply typed calculus is strongly normalising. AMS Subject Classification (1991): 03B40, 03D70. CR Subject Classification (1991): F.3.3, F.4.1. Keywords & Phrases: calculus, normalisation, perpetual strategies. Note: The research of the first author is supported by NWO/SION project 612316606. This report is also available as Computing Science Report 95/20, Eindhoven University of Technology. 1. Introduction 2 1. Introduction This paper represents an effort to shed some more light on various results concerning normalisation in calculus. We deal with calculus with only fireduction. As a first step towards a better understanding we characterise both the set of weakly normalising terms and the set of strongly norm...
0 Least Upper Bounds on the Size of Confluence and ChurchRosser Diagrams in Term Rewriting and λCalculus 1
"... We study confluence and the ChurchRosser property in term rewriting and λcalculus with explicit bounds on term sizes and reduction lengths. Given a system R, we are interested in the lengths of the reductions in the smallest valleys t → ∗ s ′ ∗ ← t ′ expressed as a function: — for confluence a ..."
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We study confluence and the ChurchRosser property in term rewriting and λcalculus with explicit bounds on term sizes and reduction lengths. Given a system R, we are interested in the lengths of the reductions in the smallest valleys t → ∗ s ′ ∗ ← t ′ expressed as a function: — for confluence a function vsR(m, n) where the valleys are for peaks t ∗ ← s → ∗ t ′ with s of size at most m and the reductions of maximum length n, and — for the ChurchRosser property a function cvsR(m, n) where the valleys are for conversions t ↔ ∗ t ′ with t and t ′ of size at most m and the conversion of maximum length n. For confluent term rewriting systems (TRSs), we prove that vsR is a total computable function, and for linear such systems that cvsR is a total computable function. Conversely, we show that every total computable function is the lower bound on the functions vsR(m, n) and cvsR(m, n) for some TRS R: In particular, we show that for every total computable function ϕ: N − → N there is a TRS R with a single term s such that vsR(s, n) ≥ ϕ(n) and cvsR(n, n) ≥ ϕ(n) for all n. For orthogonal TRSs R we prove that there is a constant k such that (a) vsR(m, n) is bounded from above by a function exponential in k and (b) cvsR(m, n) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy. Similarly, for λcalculus, we show that vsR(m, n) is bounded from above by a function in the fourth level of the Grzegorczyk hierarchy.
Parameters in Pure Type Systems
"... Abstract. In this paper we study the addition of parameters to typed�calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better finetuning of the strength of type systems as well as staying closer to type systems used in pra ..."
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Abstract. In this paper we study the addition of parameters to typed�calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better finetuning of the strength of type systems as well as staying closer to type systems used in practice in theorem provers and programming languages. 1 What are parameters? Parameters occur when functions are only allowed to occur when provided with arguments. As we will show below, both in mathematics and in programming languages the use of parameters is abundant and closely connected to the use of constants and definitions. If we want to be able to use type systems in accordance with practice and yet described in a precise manner, we therefore need parameters, constants, and definitions in type theory as well. Parameters, constants and and���� � ��������� definitions in theorem proving It is interesting to note that